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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 94d15953c5116999ac15377515b6c7e5ae430ea7
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
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commit 94d15953c5116999ac15377515b6c7e5ae430ea7
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Thu Apr 25 20:11:27 2013 -0700
Add continued fractions for erf, erfc, w and Dawson's integral.
qd-gamma.lisp::
Add continued fractions
rt-tests.lisp::
Add a test for erfc
diff --git a/qd-gamma.lisp b/qd-gamma.lisp
index 40a45eb..11e9822 100644
--- a/qd-gamma.lisp
+++ b/qd-gamma.lisp
@@ -275,6 +275,107 @@
#'(lambda (n)
(* 4 n z2)))))))
+;; From Cuyt
+;;
+;; sqrt(%pi)*z*exp(z^2)*erf(z) = K
+;;
+;; where K is the continued fraction with terms F[m]*z^2/(1 + G[m]*z^2)
+;; with F[1] = 2 and F[m] = 4*(m-1)/(2*m-3)/(2*m-1) and G[m] = -2/(2*m-1)
+;;
+(defun cf-erf (z)
+ (let ((z2 (* z z)))
+ (* (/ (exp (- z2))
+ (sqrt (float-pi z))
+ z)
+ (lentz #'(lambda (n)
+ (if (zerop n)
+ (float 0 (realpart z))
+ (1+ (/ (* -2 z2)
+ (+ n n -1)))))
+ #'(lambda (n)
+ (if (= n 1)
+ (* 2 z2)
+ (/ (* z2 4 (- n 1))
+ (* (+ n n -3) (+ n n -1)))))))))
+
+;; From the above, we also have Dawson's integral:
+;;
+;; exp(-z^-2)*integrate(exp(t^2), t, 0, z) = %i*sqrt(%pi)/2*exp(-z^2)*erf(-%i*z);
+;;
+;; -2*z*exp(-z^2)*integrate(exp(t^2), t, 0, z) = K
+;;
+;; with K = -F[m)*z^2/(1 - G[m]*z^2), where F[m] and G[m] are as above.
+;;
+;; Also erf(-%i*z) = dawson(z) * 2*exp(-z^2)/(*%i*sqrt(%pi))
+(defun cf-dawson (z)
+ (let ((z2 (* z z)))
+ (/ (lentz #'(lambda (n)
+ (if (zerop n)
+ (float 0 (realpart z))
+ (- 1 (/ (* -2 z2)
+ (+ n n -1)))))
+ #'(lambda (n)
+ (if (= n 1)
+ (* -2 z2)
+ (/ (* z2 -4 (- n 1))
+ (* (+ n n -3) (+ n n -1))))))
+ (* -2 z))))
+
+;; erfc(z) = z/sqrt(%pi)*exp(-z^2)*K
+;;
+;; where K is the continued fraction with a[1] = 1, a[m] = (m-1)/2,
+;; for m >= 2 and b[0] = 0, b[2*m+1] = z^2, b[2*m] = 1.
+;;
+;; This is valid only if Re(z) > 0.
+
+(defun cf-erfc (z)
+ (let ((z2 (* z z))
+ (zero (float 0 (realpart z)))
+ (one (float 1 (realpart z))))
+ (* (exp (- z2))
+ z
+ (/ (sqrt (float-pi (realpart z))))
+ (lentz #'(lambda (n)
+ (if (zerop n)
+ zero
+ (if (evenp n)
+ one
+ z2)))
+ #'(lambda (n)
+ (if (= n 1)
+ one
+ (/ (- n 1) 2)))))))
+
+;; w(z) = exp(-z^2)*erfc(-%i*z)
+;;
+;; = -%i*z/sqrt(%pi)*K
+;;
+;; where K is the continued fraction with a[n] the same as for erfc
+;; and b[0] = 0, b[2*m+1] = -z^2, b[2*m] = 1.
+;;
+;; This is valid only if Im(z) > 0. We can use the following
+;; identities:
+;;
+;; w(-z) = 2*exp(-z^2) - w(z)
+;; w(conj(z)) = conj(w(-z))
+
+(defun cf-w (z)
+ (let ((z2 (* z z))
+ (zero (float 0 (realpart z)))
+ (one (float 1 (realpart z))))
+ (* #c(0 -1)
+ z
+ (/ (sqrt (float-pi (realpart z))))
+ (lentz #'(lambda (n)
+ (if (zerop n)
+ zero
+ (if (evenp n)
+ one
+ (- z2))))
+ #'(lambda (n)
+ (if (= n 1)
+ one
+ (/ (- n 1) 2)))))))
;; Tail of the incomplete gamma function:
;; integrate(x^(a-1)*exp(-x), x, z, inf)
;;
diff --git a/rt-tests.lisp b/rt-tests.lisp
index af1d00c..caf568f 100644
--- a/rt-tests.lisp
+++ b/rt-tests.lisp
@@ -1729,3 +1729,8 @@
when result
append (list (list (list k x) result)))
nil)
+
+(rt:deftest erfc
+ (check-accuracy 210 (erfc #q-4)
+ #q1.9999999845827420997199811478403265131159514278547464108088316570950057869589732)
+ nil)
-----------------------------------------------------------------------
Summary of changes:
qd-gamma.lisp | 101 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rt-tests.lisp | 5 +++
2 files changed, 106 insertions(+)
hooks/post-receive
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OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 6e6d810e0e1e9064dba180f758cad56a31877f1f
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
This is an automated email from the git hooks/post-receive script. It was
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the project "OCT: A portable Lisp implementation for quad-double precision floats".
The branch, master has been updated
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- Log -----------------------------------------------------------------
commit 6e6d810e0e1e9064dba180f758cad56a31877f1f
Author: Raymond Toy <rtoy(a)google.com>
Date: Sun Nov 24 20:11:29 2013 -0800
Update the version.
diff --git a/oct.asd b/oct.asd
index 10cf0f9..1611fa8 100644
--- a/oct.asd
+++ b/oct.asd
@@ -36,7 +36,7 @@
:author "Raymond Toy"
:maintainer "See <http://www.common-lisp.net/project/oct>"
:licence "MIT"
- :version "2011.12.05" ; Just use the date
+ :version "2013.11.24" ; Just use the date
:components
((:file "qd-package")
(:file "qd-rep" :depends-on ("qd-package"))
-----------------------------------------------------------------------
Summary of changes:
oct.asd | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
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OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 0a607f3a6a6297597ae2a6c9896844803d42a6d9
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
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the project "OCT: A portable Lisp implementation for quad-double precision floats".
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via 793166a087afffde87c6d12a509069316884fd44 (commit)
via 4d09f5a39889e782bef99e2dda6d0d38327fa9f4 (commit)
via b9a21f1f536aec53c186e4a8d40d75f9d6918bb1 (commit)
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- Log -----------------------------------------------------------------
commit 0a607f3a6a6297597ae2a6c9896844803d42a6d9
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Tue Nov 26 09:55:46 2013 -0800
Switch to using atan-qd/taylor as the default atan implementation.
qd-fun.lisp:
o Use atan-qd/taylor and atan2-qd/taylor as the atan-qd and atan2-qd
implementation.
o Fix bug in atan-qd/taylor. We need a special case for x small where
we don't want to use the table. (Found by our rt-test. Yay!)
rt-tests.lisp:
o Add comment for oct.atan.4 for why it works.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index 9564dcc..207a51b 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1188,7 +1188,10 @@ is the cosine of A"
;; atan(x) = pi/2-atan(1/x), for x > 1
(sub-qd +qd-pi/2+
(atan-qd/taylor (div-qd +qd-one+ y))))
+ ((qd-<= y (aref octi::+qd-atan-partition+ 0))
+ (atan-taylor y))
(t
+ ;; The general case where the table is needed.
(let* ((i (find-atan-partition y))
(atan-xi (mul-qd +qd-pi/2+
(rational-to-qd (/ (1- (+ i 2))
@@ -1244,6 +1247,8 @@ is the cosine of A"
(neg-qd +qd-pi/4+))))
(let ((arg (atan-qd/taylor (div-qd y (abs-qd x)))))
+ ;; arg = atan2(y,|x|), so |arg| <= pi/2. Handle the case when x
+ ;; is negative.
(cond ((minusp-qd x)
(if (minusp-qd y)
(- (neg-qd +qd-pi+) arg)
@@ -1254,12 +1259,12 @@ is the cosine of A"
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
(declare (type %quad-double y x))
- (atan2-qd/newton y x))
+ (atan2-qd/taylor y x))
(defun atan-qd (y)
"Return the arc tangent of the %QUAD-DOUBLE number Y"
(declare (type %quad-double y))
- (atan-qd/newton y))
+ (atan-qd/taylor y))
(defun asin-qd (a)
"Return the arc sine of the %QUAD-DOUBLE number A"
diff --git a/rt-tests.lisp b/rt-tests.lisp
index 69f6ee7..289131e 100644
--- a/rt-tests.lisp
+++ b/rt-tests.lisp
@@ -222,6 +222,9 @@
nil)
(rt:deftest oct.atan.4
+ ;; We know atan(10^100) is pi/2 because atan(10^100) =
+ ;; pi/2-atan(10^-100) and atan(10^-100) is approx 10^-100, which
+ ;; is too small to affect pi/2.
(let* ((arg #q1q100)
(y (/ (atan arg) +pi+))
(true #q.5))
commit 793166a087afffde87c6d12a509069316884fd44
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Tue Nov 26 09:36:02 2013 -0800
Reduce size of the atan partition and node tables by half.
qd-const.lisp:
o Update tables and generator to use tables with only half as many
entries as before. This is because atan-qd/taylor never needs to
compute directly for x > 1.
qd-fun.lisp:
o Use table length for binary search, so we're independent of other
constants.
o Remove FIXME.
diff --git a/qd-const.lisp b/qd-const.lisp
index 98c5ac9..f3a5bdc 100644
--- a/qd-const.lisp
+++ b/qd-const.lisp
@@ -3770,10 +3770,15 @@
;;
;; tan((2*k + 3)*pi/(4*s))
;;
-;; for k = 0, 1, ..., s, where s is the table size,
+;; for k = 0, 1, ..., s, where s is the partition size,
;; +qd-atan-partition-size+.
+;;
+;; This table, however only contains values up to k = s/2. The
+;; algorith for atan-qd/taylor uses the relationship atan(x) =
+;; pi/2-atan(1/x) for x > 1, so we never need to compute directly
+;; atan(x) for x > 1.
(defconstant +qd-atan-partition+
- (make-array +qd-atan-partition-size+ :initial-contents
+ (make-array (/ +qd-atan-partition-size+ 2) :initial-contents
(list
(multiple-value-bind (q0 q1 q2 q3)
(renorm-5 (scale-float (float 5305687677623182 1.0d0) -61)
@@ -7358,10766 +7363,3601 @@
(scale-float (float -8606500254519528 1.0d0) -165)
(scale-float (float -6855951617059037 1.0d0) -223)
(scale-float (float 7929633697693696 1.0d0) -277))
- (%make-qd-d q0 q1 q2 q3))
- (multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4524372768344304 1.0d0) -52)
- (scale-float (float 8317904890851914 1.0d0) -107)
- (scale-float (float 8594750619195149 1.0d0) -166)
- (scale-float (float -7306789725115964 1.0d0) -220)
- (scale-float (float -8574773586034688 1.0d0) -274))
- (%make-qd-d q0 q1 q2 q3))
- (multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4538274952087405 1.0d0) -52)
- (scale-float (float 8326257921126996 1.0d0) -106)
- (scale-float (float -6022210771707739 1.0d0) -162)
- (scale-float (float -4520561563519313 1.0d0) -216)
- (scale-float (float 7859974008373248 1.0d0) -270))
- (%make-qd-d q0 q1 q2 q3))
+ (%make-qd-d q0 q1 q2 q3)))))
+
+;; This is the set of evaluation nodes for the atan table. The table contains
+;;
+;; tan((2*k + 2)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the partition size,
+;; +qd-atan-partition-size+.
+;;
+;; As explained in +qd-atan-partition+, we only store values up to k =
+;; s/2.
+(defconstant +qd-atan-nodes+
+ (make-array (/ +qd-atan-partition-size+ 2) :initial-contents
+ (list
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4552220182163332 1.0d0) -52)
- (scale-float (float -7688418407588221 1.0d0) -113)
- (scale-float (float 6056179036562030 1.0d0) -167)
- (scale-float (float 4940221467301023 1.0d0) -222)
- (scale-float (float -4749127669776384 1.0d0) -276))
+ (renorm-5 (scale-float (float 7074243300823009 1.0d0) -62)
+ (scale-float (float 6767281937885683 1.0d0) -116)
+ (scale-float (float 6495058557496386 1.0d0) -172)
+ (scale-float (float 5247049684764422 1.0d0) -226)
+ (scale-float (float -5541122207547392 1.0d0) -280))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4566208724748110 1.0d0) -52)
- (scale-float (float 6076277320393968 1.0d0) -108)
- (scale-float (float 7876705740801642 1.0d0) -162)
- (scale-float (float -6201281096292325 1.0d0) -218)
- (scale-float (float 6268449461895168 1.0d0) -272))
+ (renorm-5 (scale-float (float 7074259947269390 1.0d0) -61)
+ (scale-float (float -7016692485563189 1.0d0) -116)
+ (scale-float (float 7360987505494805 1.0d0) -170)
+ (scale-float (float 7711403976890054 1.0d0) -224)
+ (scale-float (float 7755379282149376 1.0d0) -278))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4580240848082718 1.0d0) -52)
- (scale-float (float 8179344656213134 1.0d0) -106)
- (scale-float (float 8263941875504708 1.0d0) -160)
- (scale-float (float -4889725374845191 1.0d0) -214)
- (scale-float (float -8473860704370688 1.0d0) -271))
+ (renorm-5 (scale-float (float 5305715768666702 1.0d0) -60)
+ (scale-float (float 8471749856458935 1.0d0) -117)
+ (scale-float (float -8118241448508008 1.0d0) -173)
+ (scale-float (float 5553785129213244 1.0d0) -228)
+ (scale-float (float 7351462873006080 1.0d0) -282))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4594316822493616 1.0d0) -52)
- (scale-float (float 7021650685021080 1.0d0) -107)
- (scale-float (float 5593581328329166 1.0d0) -161)
- (scale-float (float -4904011921719053 1.0d0) -217)
- (scale-float (float -5241820563505152 1.0d0) -272))
+ (renorm-5 (scale-float (float 7074326533995023 1.0d0) -60)
+ (scale-float (float -7254034420744410 1.0d0) -115)
+ (scale-float (float 6401434372892788 1.0d0) -170)
+ (scale-float (float -7987115629600412 1.0d0) -231)
+ (scale-float (float 7048491884871680 1.0d0) -285))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4608436920413514 1.0d0) -52)
- (scale-float (float 6181796558444830 1.0d0) -106)
- (scale-float (float 7403778107152819 1.0d0) -160)
- (scale-float (float -8414786340930895 1.0d0) -215)
- (scale-float (float -5307068381282304 1.0d0) -269))
+ (renorm-5 (scale-float (float 8842970593782984 1.0d0) -60)
+ (scale-float (float -5691728595114215 1.0d0) -114)
+ (scale-float (float 8352032547859414 1.0d0) -177)
+ (scale-float (float 8515194392491862 1.0d0) -232)
+ (scale-float (float -8285511068155904 1.0d0) -288))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4622601416402395 1.0d0) -52)
- (scale-float (float -7834869622036172 1.0d0) -106)
- (scale-float (float -7307397629585693 1.0d0) -163)
- (scale-float (float -6569063674381729 1.0d0) -217)
- (scale-float (float 6809893652660224 1.0d0) -275))
+ (renorm-5 (scale-float (float 5305828136410355 1.0d0) -59)
+ (scale-float (float 5670480622612706 1.0d0) -113)
+ (scale-float (float 8424458586146156 1.0d0) -167)
+ (scale-float (float 7325110716144970 1.0d0) -221)
+ (scale-float (float 7901715692986368 1.0d0) -275))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
- (renorm-5 (scale-float (float 4636810587168778 1.0d0) -52)
- (scale-float (float 6830096924257010 1.0d0) -106)
- (scale-float (float -5427308353430949 1.0d0) -161)
- (scale-float (float -7802984689345102 1.0d0) -215)
- (scale-float (float -8927666695487488 1.0d0) -269))
+ (renorm-5 (scale-float (float 6190195948340938 1.0d0) -59)
+ (scale-float (float -8802172611714431 1.0d0) -115)
+ (scale-float (float 5293066366625524 1.0d0) -169)
+ (scale-float (float -4725513813851070 1.0d0) -226)
+ (scale-float (float -6881914124238848 1.0d0) -282))
(%make-qd-d q0 q1 q2 q3))
(multiple-value-bind (q0 q1 q2 q3)
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-
-;; This is the set of evaluation nodes for the atan table. The table contains
-;;
-;; tan((2*k + 2)*pi/(4*s))
-;;
-;; for k = 0, 1, ..., s, where s is the table size,
-;; +qd-atan-partition-size+.
-(defconstant +qd-atan-nodes+
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#||
@@ -18208,24 +11048,24 @@ something that we can use to initialize our tables.
;; Create atan table
(let (val)
- (dotimes (k +qd-atan-partition-size+)
+ (dotimes (k (/ +qd-atan-partition-size+ 2))
(push (convert-to-qd-parts (tan (* pi (/ (+ 3 (* 2 k))
(* 4 +qd-atan-partition-size+)))))
val))
(write `(defconstant +qd-atan-partition+
- (make-array +qd-atan-partition-size+
+ (make-array (/ +qd-atan-partition-size+ 2)
:initial-contents (list ,@(nreverse val))))
:case :downcase
:pretty t)
(values))
(let (val)
- (dotimes (k +qd-atan-partition-size+)
+ (dotimes (k (/ +qd-atan-partition-size+ 2))
(push (convert-to-qd-parts (tan (* pi (/ (+ 2 (* 2 k))
(* 4 +qd-atan-partition-size+)))))
val))
(write `(defconstant +qd-atan-nodes+
- (make-array +qd-atan-partition-size+
+ (make-array (/ +qd-atan-partition-size+ 2)
:initial-contents (list ,@(nreverse val))))
:case :downcase
:pretty t)
diff --git a/qd-fun.lisp b/qd-fun.lisp
index 35dbcd4..9564dcc 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1117,10 +1117,8 @@ is the cosine of A"
;; the smallest entry greater than x. Return the index of the
;; entry.
(let* ((low 0)
- (high (1- octi::+qd-atan-partition-size+))
+ (high (length octi::+qd-atan-partition+))
(mid (ash (+ low high) -1)))
- ;; high starts one less than the end because the end actually has junk.
- ;; FIXME: Remove that junk last element.
(loop
(if (qd-< x (aref octi::+qd-atan-partition+ mid))
(setf high mid)
commit 4d09f5a39889e782bef99e2dda6d0d38327fa9f4
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Tue Nov 26 09:11:45 2013 -0800
Fix atan/taylor bug and add atan2/taylor.
qd-fun.lisp:
o Fix issue with atan-qd/taylor for large args. We use the
relationship atan(x) = pi/2-atan(1/x) for x > 1.
o Add atan2-qd/taylor. This appears to be much faster than
atan2-qd/newton.
timing.lisp:
o Add timing test for atan2-qd.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index e782a8a..35dbcd4 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1182,21 +1182,76 @@ is the cosine of A"
;; atan(x) = (2*i-1)*pi/(4*s) + atan(e)
;;
;; Note that qd-atan-partition array starts with X[2].
- (if (minusp-qd y)
- (neg-qd (atan-qd/taylor (neg-qd y)))
- (let* ((i (find-atan-partition y))
- (atan-xi (mul-qd +qd-pi/2+
- (rational-to-qd (/ (1- (+ i 2))
- octi::+qd-atan-partition-size+))))
- (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
- (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
- (add-qd 1/xi y)))
- (e (sub-qd 1/xi
- e1)))
- ;;(format t "partition ~D~%" i)
- ;;(format t "atan(xi) = ~S~%" atan-xi)
- ;;(format t "e = ~S~%" e)
- (add-qd atan-xi (atan-taylor e)))))
+ (cond
+ ((minusp-qd y)
+ ;; atan(-x) = -atan(x)
+ (neg-qd (atan-qd/taylor (neg-qd y))))
+ ((qd-> y +qd-one+)
+ ;; atan(x) = pi/2-atan(1/x), for x > 1
+ (sub-qd +qd-pi/2+
+ (atan-qd/taylor (div-qd +qd-one+ y))))
+ (t
+ (let* ((i (find-atan-partition y))
+ (atan-xi (mul-qd +qd-pi/2+
+ (rational-to-qd (/ (1- (+ i 2))
+ octi::+qd-atan-partition-size+))))
+ (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
+ (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
+ (add-qd 1/xi y)))
+ (e (sub-qd 1/xi
+ e1)))
+ ;;(format t "partition ~D~%" i)
+ ;;(format t "atan(xi) = ~S~%" atan-xi)
+ ;;(format t "e = ~S~%" e)
+ (add-qd atan-xi (atan-taylor e))))))
+
+(defun atan2-qd/taylor (y x)
+ (declare (type %quad-double y x)
+ #+nil
+ (optimize (speed 3) (space 0)))
+ (cond ((zerop-qd x)
+ ;; x = 0
+ (cond ((zerop-qd y)
+ ;; Both x and y are zero. Use the signs of x and y to
+ ;; determine the result
+ (error "atan2(0,0)"))
+ (t
+ ;; x = 0, but y is not. Use the sign of y.
+ (return-from atan2-qd/taylor
+ (cond ((plusp (float-sign (qd-0 y)))
+ +qd-pi/2+)
+ (t
+ (neg-qd +qd-pi/2+)))))))
+ ((zerop-qd y)
+ ;; y = 0.
+ (return-from atan2-qd/taylor
+ ;; Use the sign of x and y to figure out the result.
+ (cond ((plusp (float-sign (qd-0 x)))
+ +qd-zero+)
+ ((plusp (float-sign (qd-0 y)))
+ +qd-pi+)
+ (t
+ (neg-qd +qd-pi+))))))
+
+ (when (qd-= x y)
+ (return-from atan2-qd/taylor
+ (if (plusp-qd y)
+ +qd-pi/4+
+ +qd-3pi/4+)))
+
+ (when (qd-= x (neg-qd y))
+ (return-from atan2-qd/taylor
+ (if (plusp-qd y)
+ +qd-3pi/4+
+ (neg-qd +qd-pi/4+))))
+
+ (let ((arg (atan-qd/taylor (div-qd y (abs-qd x)))))
+ (cond ((minusp-qd x)
+ (if (minusp-qd y)
+ (- (neg-qd +qd-pi+) arg)
+ (- +qd-pi+ arg)))
+ (t
+ arg))))
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
diff --git a/timing.lisp b/timing.lisp
index 1847c4c..9e2074a 100644
--- a/timing.lisp
+++ b/timing.lisp
@@ -231,3 +231,57 @@
;; We see that the taylor series is 21 times faster (!) and conses 11
;; times less. That's a pretty nice gain, at the expense of two
;; 1024-element tables.
+
+(defun time-atan2 (&optional (n 10000))
+ (declare (fixnum n))
+ (flet ((time-atan/newton ()
+ (let ((sum (octi::make-qd-d -1000d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan2-qd/newton sum (mul-qd-d sum .1d0))))
+ sum))
+ (time-atan/taylor ()
+ (let ((sum (octi::make-qd-d -1000d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan2-qd/taylor sum (mul-qd-d sum .1d0))))
+ sum)))
+ (format t "atan2-qd/newton ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/newton)))
+ (format t "atan2-qd/taylor ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/taylor)))))
+
+
+;; Some timing results on an iMac, 3.06 GHz Core i3:
+
+; atan2-qd/newton 10000 times
+
+; Evaluation took:
+; 6.37 seconds of real time
+; 6.321544 seconds of user run time
+; 0.026819 seconds of system run time
+; 19,474,336,804 CPU cycles
+; [Run times include 0.27 seconds GC run time]
+; 0 page faults and
+; 934,872,144 bytes consed.
+;
+; sum = #C(-1.670464979286058652105921398771024w0 1.39712341620073544419882362664172w-33)
+;
+; atan2-qd/taylor 10000 times
+
+; Evaluation took:
+; 0.32 seconds of real time
+; 0.320343 seconds of user run time
+; 0.001606 seconds of system run time
+; 989,558,772 CPU cycles
+; [Run times include 0.02 seconds GC run time]
+; 0 page faults and
+; 88,919,384 bytes consed.
+;
+; sum = #C(-1.670464979286058652105921398771027w0 4.47861132722031280908838833477765w-33)
commit b9a21f1f536aec53c186e4a8d40d75f9d6918bb1
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Mon Nov 25 23:10:26 2013 -0800
Implement a new atan method that is 21 times faster.
The new method uses a table and the taylor series for atan to compute
atan for real args.
qd-const.lisp:
o Add the two tables used for the atan computation.
o Include clisp code that generates the tables.
qd-fun.lisp:
o Implement atan using a set of tables and a Taylor series.
timing.lisp:
o Add a simple timing program to evaluate how fast each atan method
is.
diff --git a/qd-const.lisp b/qd-const.lisp
index 56840b7..98c5ac9 100644
--- a/qd-const.lisp
+++ b/qd-const.lisp
@@ -3762,6 +3762,14363 @@
(%make-qd-d q0 q1 q2 q3))
"Quad-double value of Euler's constant")
+;; Table for atan computation. See atan-qd/taylor.
+(defconstant +qd-atan-partition-size+
+ 1024)
+
+;; This is the partition points for the atan table. The table contains
+;;
+;; tan((2*k + 3)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the table size,
+;; +qd-atan-partition-size+.
+(defconstant +qd-atan-partition+
+ (make-array +qd-atan-partition-size+ :initial-contents
+ (list
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5305687677623182 1.0d0) -61)
+ (scale-float (float 5506082357036760 1.0d0) -115)
+ (scale-float (float -8811919955122674 1.0d0) -170)
+ (scale-float (float 6519548665950046 1.0d0) -226)
+ (scale-float (float -6440164966137856 1.0d0) -280))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8842840540207337 1.0d0) -61)
+ (scale-float (float -5473160723154885 1.0d0) -115)
+ (scale-float (float -4535918610883809 1.0d0) -171)
+ (scale-float (float -4963578518408804 1.0d0) -225)
+ (scale-float (float -6712212661567488 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6190017509796469 1.0d0) -60)
+ (scale-float (float 8647620749142769 1.0d0) -115)
+ (scale-float (float -6931221986832039 1.0d0) -171)
+ (scale-float (float -8517197728414346 1.0d0) -225)
+ (scale-float (float -4626619940470784 1.0d0) -283))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7958643881798693 1.0d0) -60)
+ (scale-float (float -5354492433078344 1.0d0) -114)
+ (scale-float (float -7327905104279995 1.0d0) -168)
+ (scale-float (float -6523976005184669 1.0d0) -223)
+ (scale-float (float -7854698297393152 1.0d0) -277))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4863653855283737 1.0d0) -59)
+ (scale-float (float -6015787009420778 1.0d0) -113)
+ (scale-float (float 8594253106416801 1.0d0) -167)
+ (scale-float (float 7046824309216708 1.0d0) -222)
+ (scale-float (float -6520970423566336 1.0d0) -277))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5748008660632557 1.0d0) -59)
+ (scale-float (float 8797928203970591 1.0d0) -113)
+ (scale-float (float 8714055176617153 1.0d0) -167)
+ (scale-float (float -5612171940322240 1.0d0) -221)
+ (scale-float (float 4939191171620864 1.0d0) -275))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6632390519957926 1.0d0) -59)
+ (scale-float (float 8670207667135100 1.0d0) -114)
+ (scale-float (float 8149147453636330 1.0d0) -168)
+ (scale-float (float -8078741638624220 1.0d0) -222)
+ (scale-float (float 5251828477165568 1.0d0) -277))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7516803596781324 1.0d0) -59)
+ (scale-float (float -6201171919502881 1.0d0) -116)
+ (scale-float (float -7027033891914616 1.0d0) -173)
+ (scale-float (float 5952967054329307 1.0d0) -227)
+ (scale-float (float -8689209012387840 1.0d0) -282))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8401252055212014 1.0d0) -59)
+ (scale-float (float -8018416525424526 1.0d0) -114)
+ (scale-float (float -8131489574139489 1.0d0) -168)
+ (scale-float (float 5356676700338153 1.0d0) -223)
+ (scale-float (float -8760547365355520 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4642870030012741 1.0d0) -58)
+ (scale-float (float 8661794336121368 1.0d0) -112)
+ (scale-float (float 5441429837488339 1.0d0) -167)
+ (scale-float (float -6919382284775521 1.0d0) -222)
+ (scale-float (float -5506479342911488 1.0d0) -276))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5085135888370956 1.0d0) -58)
+ (scale-float (float -7882846098076715 1.0d0) -113)
+ (scale-float (float -7762714374541670 1.0d0) -171)
+ (scale-float (float 8931845445931377 1.0d0) -225)
+ (scale-float (float 5556186022150144 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5527425685852335 1.0d0) -58)
+ (scale-float (float 8127719015560079 1.0d0) -112)
+ (scale-float (float -6094331376989970 1.0d0) -166)
+ (scale-float (float 6973207316824032 1.0d0) -220)
+ (scale-float (float -8483611529240576 1.0d0) -274))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5969741506079427 1.0d0) -58)
+ (scale-float (float 8989322454423522 1.0d0) -112)
+ (scale-float (float 4820491773276110 1.0d0) -171)
+ (scale-float (float 5078156230612779 1.0d0) -225)
+ (scale-float (float 6795293461839872 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6412085433164924 1.0d0) -58)
+ (scale-float (float 5647386557728108 1.0d0) -115)
+ (scale-float (float 5565185291589407 1.0d0) -169)
+ (scale-float (float -8187312701179060 1.0d0) -225)
+ (scale-float (float -8291776467828736 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6854459551750966 1.0d0) -58)
+ (scale-float (float -5878879597999219 1.0d0) -112)
+ (scale-float (float -4586089337686870 1.0d0) -166)
+ (scale-float (float -8194332440371855 1.0d0) -224)
+ (scale-float (float 6231038519017472 1.0d0) -278))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7296865947048466 1.0d0) -58)
+ (scale-float (float 5117941371137976 1.0d0) -114)
+ (scale-float (float 4515350251087483 1.0d0) -169)
+ (scale-float (float 5446926177260784 1.0d0) -223)
+ (scale-float (float 6233667346104320 1.0d0) -280))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7739306704876459 1.0d0) -58)
+ (scale-float (float 5788179125278216 1.0d0) -112)
+ (scale-float (float -8864296878521864 1.0d0) -169)
+ (scale-float (float -6359044574484051 1.0d0) -225)
+ (scale-float (float 6661760607584256 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8181783911701472 1.0d0) -58)
+ (scale-float (float -6390306030927826 1.0d0) -112)
+ (scale-float (float 8314328435451015 1.0d0) -167)
+ (scale-float (float 5656079407037522 1.0d0) -221)
+ (scale-float (float 8659304386199552 1.0d0) -276))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 8624299654676919 1.0d0) -58)
+ (scale-float (float 5636124989072738 1.0d0) -115)
+ (scale-float (float -6628018644317178 1.0d0) -169)
+ (scale-float (float 7868931088360995 1.0d0) -225)
+ (scale-float (float -5989342687264768 1.0d0) -281))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4533428010841266 1.0d0) -57)
+ (scale-float (float 7873841147516662 1.0d0) -112)
+ (scale-float (float -6907345987993589 1.0d0) -166)
+ (scale-float (float -7970290490656245 1.0d0) -220)
+ (scale-float (float 8183217733074944 1.0d0) -274))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4754727550681906 1.0d0) -57)
+ (scale-float (float -6794493206896267 1.0d0) -116)
+ (scale-float (float 6762402539062604 1.0d0) -170)
+ (scale-float (float 8891551854319655 1.0d0) -224)
+ (scale-float (float -6933069732249600 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4976049491585756 1.0d0) -57)
+ (scale-float (float -7056882828989748 1.0d0) -115)
+ (scale-float (float 7164953570628209 1.0d0) -169)
+ (scale-float (float 7677417338552630 1.0d0) -224)
+ (scale-float (float -5298942914330624 1.0d0) -279))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5197394878700579 1.0d0) -57)
+ (scale-float (float 6124273349739532 1.0d0) -115)
+ (scale-float (float -8600581353254220 1.0d0) -170)
+ (scale-float (float -6717554843753204 1.0d0) -224)
+ (scale-float (float 5322629696192512 1.0d0) -278))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5418764757616299 1.0d0) -57)
+ (scale-float (float 7551546303483781 1.0d0) -112)
+ (scale-float (float 5433875429936579 1.0d0) -173)
+ (scale-float (float -7130504460297287 1.0d0) -227)
+ (scale-float (float -8382177370701824 1.0d0) -281))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5640160174384795 1.0d0) -57)
+ (scale-float (float 5342817502011044 1.0d0) -112)
+ (scale-float (float -5198387028085064 1.0d0) -167)
+ (scale-float (float -4984834528418682 1.0d0) -222)
+ (scale-float (float 6854185795649536 1.0d0) -276))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5861582175539712 1.0d0) -57)
+ (scale-float (float -8030596855746236 1.0d0) -112)
+ (scale-float (float 7403724276832834 1.0d0) -167)
+ (scale-float (float -7591693793173951 1.0d0) -223)
+ (scale-float (float 7380449510424576 1.0d0) -277))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6083031808116292 1.0d0) -57)
+ (scale-float (float -8939232852017447 1.0d0) -111)
+ (scale-float (float 7615734983381517 1.0d0) -166)
+ (scale-float (float -8294165265443062 1.0d0) -220)
+ (scale-float (float 4657370779582464 1.0d0) -275))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
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+ (%make-qd-d q0 q1 q2 q3))
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+
+;; This is the set of evaluation nodes for the atan table. The table contains
+;;
+;; tan((2*k + 2)*pi/(4*s))
+;;
+;; for k = 0, 1, ..., s, where s is the table size,
+;; +qd-atan-partition-size+.
+(defconstant +qd-atan-nodes+
+ (make-array +qd-atan-partition-size+ :initial-contents
+ (list
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+ (scale-float (float -7941956443494165 1.0d0) -210)
+ (scale-float (float 8699025989009408 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4586969091394693 1.0d0) -46)
+ (scale-float (float -8978492069766178 1.0d0) -101)
+ (scale-float (float 7053055367607091 1.0d0) -155)
+ (scale-float (float -8712227340191945 1.0d0) -210)
+ (scale-float (float -7846634958946304 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5096708286733636 1.0d0) -46)
+ (scale-float (float -4572705006895138 1.0d0) -100)
+ (scale-float (float 4718547123658870 1.0d0) -154)
+ (scale-float (float 7157224158707570 1.0d0) -210)
+ (scale-float (float -8590740628766720 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5733873284861593 1.0d0) -46)
+ (scale-float (float -8604953363186775 1.0d0) -103)
+ (scale-float (float 8626935765066666 1.0d0) -158)
+ (scale-float (float -4858320692337793 1.0d0) -212)
+ (scale-float (float -7449309035233280 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 6553075144281224 1.0d0) -46)
+ (scale-float (float 6620952143443553 1.0d0) -101)
+ (scale-float (float -5268353655146108 1.0d0) -156)
+ (scale-float (float 5893295495507489 1.0d0) -211)
+ (scale-float (float -4800425745186816 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7645332295807713 1.0d0) -46)
+ (scale-float (float -5080909471366464 1.0d0) -102)
+ (scale-float (float 7938109188081450 1.0d0) -156)
+ (scale-float (float 4928452855514440 1.0d0) -210)
+ (scale-float (float 8885897390456832 1.0d0) -268))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 4587238957440646 1.0d0) -45)
+ (scale-float (float -7194218336984116 1.0d0) -99)
+ (scale-float (float -8985364386653513 1.0d0) -153)
+ (scale-float (float -8164942491672346 1.0d0) -207)
+ (scale-float (float 6286694831915008 1.0d0) -262))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734089176174276 1.0d0) -45)
+ (scale-float (float -7647633804412292 1.0d0) -101)
+ (scale-float (float -7089963492055907 1.0d0) -155)
+ (scale-float (float 8957378262525840 1.0d0) -210)
+ (scale-float (float 8460149900050432 1.0d0) -265))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7645494213403192 1.0d0) -45)
+ (scale-float (float 4683660413726437 1.0d0) -99)
+ (scale-float (float -7150841523991633 1.0d0) -153)
+ (scale-float (float -4654257076122560 1.0d0) -207)
+ (scale-float (float 5269682304483328 1.0d0) -261))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734143148494430 1.0d0) -44)
+ (scale-float (float 8353770598111665 1.0d0) -99)
+ (scale-float (float -8962023885528779 1.0d0) -158)
+ (scale-float (float -5933825828214605 1.0d0) -212)
+ (scale-float (float 5439908491034624 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 5734156641542718 1.0d0) -43)
+ (scale-float (float 8633141086175409 1.0d0) -98)
+ (scale-float (float 5419442444201815 1.0d0) -153)
+ (scale-float (float -8712131584355876 1.0d0) -211)
+ (scale-float (float -5157997961543680 1.0d0) -266))
+ (%make-qd-d q0 q1 q2 q3))
+ (multiple-value-bind (q0 q1 q2 q3)
+ (renorm-5 (scale-float (float 7091480910882704 1.0d0) 204)
+ (scale-float (float 8415670760171201 1.0d0) 148)
+ (scale-float (float -8191062761010691 1.0d0) 94)
+ (scale-float (float 8061527243862424 1.0d0) 39)
+ (scale-float (float -8684710553714688 1.0d0) -17))
+ (%make-qd-d q0 q1 q2 q3)))))
#||
Some sample test code to convert long-float data from clisp to
@@ -3848,4 +18205,30 @@ something that we can use to initialize our tables.
(dotimes (k 256)
(push (convert-to-qd-parts (cos (* pi (/ (1+ k) 1024)))) val))
`(make-array 256 :initial-contents (list ,@(nreverse val))))
+
+;; Create atan table
+(let (val)
+ (dotimes (k +qd-atan-partition-size+)
+ (push (convert-to-qd-parts (tan (* pi (/ (+ 3 (* 2 k))
+ (* 4 +qd-atan-partition-size+)))))
+ val))
+ (write `(defconstant +qd-atan-partition+
+ (make-array +qd-atan-partition-size+
+ :initial-contents (list ,@(nreverse val))))
+ :case :downcase
+ :pretty t)
+ (values))
+
+(let (val)
+ (dotimes (k +qd-atan-partition-size+)
+ (push (convert-to-qd-parts (tan (* pi (/ (+ 2 (* 2 k))
+ (* 4 +qd-atan-partition-size+)))))
+ val))
+ (write `(defconstant +qd-atan-nodes+
+ (make-array +qd-atan-partition-size+
+ :initial-contents (list ,@(nreverse val))))
+ :case :downcase
+ :pretty t)
+ (values))
+
||#
diff --git a/qd-fun.lisp b/qd-fun.lisp
index b3ae7b6..e782a8a 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -1102,6 +1102,102 @@ is the cosine of A"
#+nil (optimize (speed 3) (space 0)))
(atan2-qd/newton y +qd-one+))
+;; Computation of atan using a table and taylor series.
+
+(defun find-atan-partition-linear (x)
+ ;; Simple linear search. Search through the qd-atan-partition table
+ ;; to find the entry k such that x > X[k]. Return k.
+ (loop for k from 0 below octi::+qd-atan-partition-size+
+ ;; do (format t "k = ~D, atan = ~S~%" k (aref octi::+qd-atan-partition+ k))
+ when (qd-< x (aref octi::+qd-atan-partition+ k))
+ return k))
+
+(defun find-atan-partition (x)
+ ;; Perform binary search through the atan partition table to find
+ ;; the smallest entry greater than x. Return the index of the
+ ;; entry.
+ (let* ((low 0)
+ (high (1- octi::+qd-atan-partition-size+))
+ (mid (ash (+ low high) -1)))
+ ;; high starts one less than the end because the end actually has junk.
+ ;; FIXME: Remove that junk last element.
+ (loop
+ (if (qd-< x (aref octi::+qd-atan-partition+ mid))
+ (setf high mid)
+ (setf low mid))
+ (setf mid (ash (+ low high) -1))
+ ;;(format t "low, mid, high = ~A ~A ~A~%" low mid high)
+ (when (<= (- high low) 1)
+ (return)))
+ ;;(format t "mid = ~A~%" mid)
+ ;;(format t "X[mid] = ~A~%" (aref octi::+qd-atan-partition+ mid))
+ (if (qd-< x (aref octi::+qd-atan-partition+ mid))
+ mid
+ (1+ mid))))
+
+
+(defun atan-taylor (a)
+ ;; Taylor series for atan(x).
+ ;;
+ ;; atan(x) = sum((-1)^k*x^(2*k+1)/(2*k+1), k, 0, inf)
+ ;; = x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - ...
+ ;; = x*(1 - x^2/3 + x^4/5 - x^6/7 + x^8/9 - ...)
+ ;;
+ ;; x should be small for this to be effective, but no check is made
+ ;; for this. The series on converges for |x| < 1.
+ (let* ((x2 (neg-qd (sqr-qd a)))
+ (term x2)
+ (s (make-qd-d 1d0))
+ (m 3))
+ (loop
+ (setf s (add-qd s (div-qd-d term (float m 1d0))))
+ (setf term (mul-qd term x2))
+ (setf m (+ m 2))
+ (when (<= (abs (qd-0 term)) +qd-eps+)
+ ;;(format t "terms = ~A~%" m)
+ (return)))
+ (mul-qd s a)))
+
+(defun atan-qd/taylor (y)
+ ;; To compute atan(x) for x in [0,inf], we partition the interval
+ ;; [0,inf] such that partition points X[i] and the evaluation nodes
+ ;; x[i] are defined by
+ ;;
+ ;; x[i] = tan((2*i-2)*pi/(4*s)), i = 2, ..., s + 1
+ ;; X[i] = tan((2*i-1)*pi/(4*s)), i = 1, ..., s
+ ;;
+ ;; where 0 = X[0] < X[1] < x[2] < X[2] < ... < x[s] < X[s] < x[s+1]
+ ;; = X[s+1] = inf.
+ ;;
+ ;; So if x is in the interval [X[i-1], X[i]}, the computation of
+ ;; atan(x) is given by
+ ;;
+ ;; atan(x) = atan(x[i]) + atan(e)
+ ;;
+ ;; where e = 1/x[i] - (1+1/x[i]^2)/(x+1/x[i])
+ ;;
+ ;; But since x[i] = tan((2*i-1)*pi/(4*s)), atan(x[i]) =
+ ;; (2*i-1)*pi/(4*s), so
+ ;;
+ ;; atan(x) = (2*i-1)*pi/(4*s) + atan(e)
+ ;;
+ ;; Note that qd-atan-partition array starts with X[2].
+ (if (minusp-qd y)
+ (neg-qd (atan-qd/taylor (neg-qd y)))
+ (let* ((i (find-atan-partition y))
+ (atan-xi (mul-qd +qd-pi/2+
+ (rational-to-qd (/ (1- (+ i 2))
+ octi::+qd-atan-partition-size+))))
+ (1/xi (div-qd +qd-one+ (aref octi::+qd-atan-nodes+ i)))
+ (e1 (div-qd (add-qd-d (sqr-qd 1/xi) 1d0)
+ (add-qd 1/xi y)))
+ (e (sub-qd 1/xi
+ e1)))
+ ;;(format t "partition ~D~%" i)
+ ;;(format t "atan(xi) = ~S~%" atan-xi)
+ ;;(format t "e = ~S~%" e)
+ (add-qd atan-xi (atan-taylor e)))))
+
(defun atan2-qd (y x)
"atan2(y, x) = atan(y/x), but carefully handling the quadrant"
(declare (type %quad-double y x))
diff --git a/timing.lisp b/timing.lisp
index 582671b..1847c4c 100644
--- a/timing.lisp
+++ b/timing.lisp
@@ -174,3 +174,60 @@
(format t "Sqrt QD-REAL (method) ~d times~%" n)
#+cmu (ext:gc :full t)
(time (sqrt-qd-real))))
+
+(defun time-atan (&optional (n 10000))
+ (declare (fixnum n))
+ (flet ((time-atan/newton ()
+ (let ((sum (octi::make-qd-d 0.01d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan-qd/newton sum)))
+ sum))
+ (time-atan/taylor ()
+ (let ((sum (octi::make-qd-d 0.01d0)))
+ (declare (type octi::%quad-double sum)
+ (optimize (speed 3)))
+ (dotimes (k n)
+ (declare (fixnum k))
+ (setf sum (octi::atan-qd/taylor sum)))
+ sum)))
+ (format t "atan-qd/newton ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/newton)))
+ (format t "atan-qd/taylor ~d times~%" n)
+ #+cmu (ext:gc :full t)
+ (format t "sum = ~A~%" (time (time-atan/taylor)))))
+
+;; Some timing results on an iMac, 3.06 GHz Core i3:
+
+; atan-qd/newton 10000 times
+
+; Evaluation took:
+; 6.15 seconds of real time
+; 6.116626 seconds of user run time
+; 0.021624 seconds of system run time
+; 18,805,071,897 CPU cycles
+; [Run times include 0.24 seconds GC run time]
+; 0 page faults and
+; 893,037,112 bytes consed.
+;
+; sum = #C(0.0077459785628163552318041722744135w0 5.8166227464838760117515984152653w-36)
+;
+; atan-qd/taylor 10000 times
+
+; Evaluation took:
+; 0.29 seconds of real time
+; 0.28987 seconds of user run time
+; 0.001629 seconds of system run time
+; 892,698,785 CPU cycles
+; [Run times include 0.02 seconds GC run time]
+; 0 page faults and
+; 79,108,384 bytes consed.
+;
+; sum = #C(0.0077459785628163552318041722744135w0 5.81662274648387601175165266362535w-36)
+
+;; We see that the taylor series is 21 times faster (!) and conses 11
+;; times less. That's a pretty nice gain, at the expense of two
+;; 1024-element tables.
-----------------------------------------------------------------------
Summary of changes:
qd-const.lisp | 7223 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
qd-fun.lisp | 158 +-
rt-tests.lisp | 3 +
timing.lisp | 111 +
4 files changed, 7493 insertions(+), 2 deletions(-)
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OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. b1d9be467f5f44383a356a4a11a04a64f0f29402
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
This is an automated email from the git hooks/post-receive script. It was
generated because a ref change was pushed to the repository containing
the project "OCT: A portable Lisp implementation for quad-double precision floats".
The branch, master has been updated
via b1d9be467f5f44383a356a4a11a04a64f0f29402 (commit)
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- Log -----------------------------------------------------------------
commit b1d9be467f5f44383a356a4a11a04a64f0f29402
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Wed Nov 27 09:09:50 2013 -0800
Add implementation for tan(x) using duplication.
Includes Lentz' algorithm for evaluating continued fractions and
updates the timing test to include tan(x) using duplication. Tests
show this is much slower than existing algorithms.
diff --git a/qd-extra.lisp b/qd-extra.lisp
index 61ce7e4..3670be5 100644
--- a/qd-extra.lisp
+++ b/qd-extra.lisp
@@ -993,6 +993,66 @@
y (add-qd (mul-qd y ct) (mul-qd x st)))
(div-qd y x))))
+(defvar *debug-cf-eval* nil)
+(defvar *max-cf-iterations* 10000)
+
+(defun lentz-qd (bf af)
+ (let ((tiny-value-count 0))
+ (flet ((value-or-tiny (v)
+ (cond ((zerop-qd v)
+ (incf tiny-value-count)
+ (make-qd-d (sqrt least-positive-normalized-double-float)))
+ (t
+ v))))
+ (let* ((f (value-or-tiny (funcall bf 0)))
+ (c f)
+ (d (make-qd-d 0d0))
+ (eps +qd-eps+))
+ (loop
+ for j from 1 upto *max-cf-iterations*
+ for an = (funcall af j)
+ for bn = (funcall bf j)
+ do (progn
+ (setf d (value-or-tiny (add-qd bn (mul-qd an d))))
+ (setf c (value-or-tiny (add-qd bn (div-qd an c))))
+ (when *debug-cf-eval*
+ (format t "~&j = ~d~%" j)
+ (format t " an = ~s~%" an)
+ (format t " bn = ~s~%" bn)
+ (format t " c = ~s~%" c)
+ (format t " d = ~s~%" d))
+ (let ((delta (div-qd c d)))
+ (setf d (div-qd +qd-one+ d))
+ (setf f (mul-qd f delta))
+ (when *debug-cf-eval*
+ (format t " dl= ~S (|dl - 1| = ~S)~%" delta (abs (1- delta)))
+ (format t " f = ~S~%" f))
+ (when (qd-<= (abs-qd (sub-qd-d delta 1d0)) eps)
+ (return-from lentz-qd (values f j tiny-value-count)))))
+ finally
+ (error 'simple-error
+ :format-control "~<Continued fraction failed to converge after ~D iterations.~% Delta = ~S~>"
+ :format-arguments (list *max-cf-iterations* (/ c d))))))))
+
+(defun tan-qd/duplication (x)
+ ;; tan(x) = 2*tan(x/2)/(1-tan(x/2)^2)
+ (cond ((< (abs (qd-0 x)) 1d-4)
+ ;; Continued fraction for tan(x)
+ ;;
+ ;; x
+ ;; tan(x) = -----------------------------
+ ;; 1 - K(2*k+1, -z^2, k, 0, inf)
+ (let ((x2 (neg-qd (sqr-qd x))))
+ (div-qd x
+ (lentz-qd #'(lambda (n)
+ (octi::make-qd-d (float (+ (* 2 n) 1) 1d0)))
+ #'(lambda (n)
+ (declare (ignore n))
+ x2)))))
+ (t
+ (let* ((tanx/2 (tan-qd/duplication (scale-float-qd x -1))))
+ (/ (mul-qd-d tanx/2 2d0)
+ (sub-d-qd 1d0 (sqr-qd tanx/2)))))))
(defun sin-qd/cordic (r)
(declare (type %quad-double r))
@@ -1374,17 +1434,22 @@
(format t "atan2-qd/newton~%")
(time (dotimes (k n)
(declare (fixnum k))
- (setf y (atan2-qd/newton x one))))
+ (setf y (octi::atan2-qd/newton x one))))
#+cmu (ext:gc :full t)
(format t "atan2-qd/cordic~%")
(time (dotimes (k n)
(declare (fixnum k))
- (setf y (atan2-qd/cordic x one))))
+ (setf y (octi::atan2-qd/cordic x one))))
#+cmu (ext:gc :full t)
(format t "atan-qd/duplication~%")
(time (dotimes (k n)
(declare (fixnum k))
- (setf y (atan-qd/duplication x))))
+ (setf y (octi::atan-qd/duplication x))))
+ #+cmu (ext:gc :full t)
+ (format t "atan-qd/taylor~%")
+ (time (dotimes (k n)
+ (declare (fixnum k))
+ (setf y (octi::atan-qd/taylor x))))
))
;; (time-tan #c(10w0 0) 10000)
-----------------------------------------------------------------------
Summary of changes:
qd-extra.lisp | 71 ++++++++++++++++++++++++++++++++++++++++++++++++++++++---
1 file changed, 68 insertions(+), 3 deletions(-)
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OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 6127a79e0d6300bc7ac7754190fbd559d00e7401
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
This is an automated email from the git hooks/post-receive script. It was
generated because a ref change was pushed to the repository containing
the project "OCT: A portable Lisp implementation for quad-double precision floats".
The branch, master has been updated
via 6127a79e0d6300bc7ac7754190fbd559d00e7401 (commit)
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Those revisions listed above that are new to this repository have
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- Log -----------------------------------------------------------------
commit 6127a79e0d6300bc7ac7754190fbd559d00e7401
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Sun Nov 24 10:41:27 2013 -0800
Fix ticket:5
o Implement better pi reduction and also always do pi reduction.
o Add test for sin(pi) and cos(2^120), to test pi reduction.
o Accuracy for erfc is only 198 bits.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index d131a7a..91c0b32 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -696,38 +696,46 @@ is the cosine of A"
(values mod f)
(values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+))))))
-(defun rem-pi/2-int (qd)
- ;; Compute qd rem pi/2 = k*pi/2+y. So we compute k + y*2/pi =
- ;; qd*2/pi.
- ;;
- ;; First convert qd to 2^e*I. We already have 2/pi in the form
- ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the
- ;; integer and fractional parts of this. For the integer part we
- ;; only need it modulo 4, because of the periodicity. For the
- ;; fractional part, we only need 212 (or so bits of fraction).
- ;;
- ;; FIXME: But we don't really need to compute all the bits of I*J.
- ;; In the product, we really only need the 2 bits to the left of the
- ;; binary point, and then 212 bits to the right. This doesn't
- ;; require doing the full multiplication.
- (multiple-value-bind (i e s)
+(defun split-prod-2/pi (qd)
+ ;; Compute k and f such that k + f = qd*2/pi, where k is an intege
+ ;; and f < 1. Since we do not need more than the the least
+ ;; two bits from k, only the least two bits of k are returned.
+ (multiple-value-bind (i e)
(integer-decode-qd qd)
+ ;; First convert qd to 2^e*I. We already have 2/pi in the form
+ ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the
+ ;; integer and fractional parts of this. For the integer part we
+ ;; only need it modulo 4, because of the periodicity. For the
+ ;; fractional bits, we return the actual fraction as a ratio.
+ ;;
+ ;; FIXME: We compute the entire product I*J, but we only need the
+ ;; 2 bits to the left of the binary point. But we need many of
+ ;; the bits to the right of the binary point becaue we have no
+ ;; lower bound on the difference between a quad-double and pi.
(let* ((exp (- e (integer-length +2/pi-bits+)))
- (prod (* (* s i) +2/pi-bits+))
- (mod (ldb (byte 2 (- exp)) prod))
- ;; A quad-double has about 212 bits, but we add another 53
- ;; (5 doubles) for some extra accuracty.
- (qd-bits 265)
- (frac (ldb (byte qd-bits (- (- exp) qd-bits)) prod))
- (f (mul-qd (scale-float-qd (rational-to-qd frac) (- qd-bits))
- +qd-pi/2+)))
- ;; We want the remainder part to be <= pi/4 because the trig
- ;; functions want that. So if the fraction is too big, adjust
- ;; it, and mod value.
- (if (qd-<= (abs-qd f) +qd-pi/4+)
- (values mod f)
- (values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+))))))
+ (binary-point (- exp))
+ (prod (* i +2/pi-bits+))
+ (int (ldb (byte 2 binary-point) prod))
+ (frac (/ (ldb (byte binary-point 0) prod)
+ (ash 1 binary-point))))
+ (values int frac))))
+(defun rem-pi/2-int (qd)
+ "Compute qd rem pi/2 = k + f, where k is an integer and |f| <
+ pi/4. Two values are returned: k mod 4 and f."
+ (if (minusp-qd qd)
+ (multiple-value-bind (k f)
+ (rem-pi/2-int (neg-qd qd))
+ (values (mod (- k) 4) (neg-qd f)))
+ (multiple-value-bind (k f)
+ (split-prod-2/pi qd)
+ ;; The trig functions want the arg to be less than pi/4. That
+ ;; is f < 1/2. Thus adjust k and f accordingly.
+ (when (> f 1/2)
+ (setf k (mod (1+ k) 4))
+ (setf f (- f 1)))
+ (values k (mul-qd +qd-pi/2+ (rational-to-qd f))))))
+
(defun rem-pi/2-int-b (qd)
(declare (type %quad-double qd))
(multiple-value-bind (i e s)
@@ -785,6 +793,13 @@ is the cosine of A"
(sub-qd a (mul-qd-d +qd-pi/2+ (float quot 1d0))))))
(t
(rem-pi/2-int a))))
+
+(defun rem-pi/2 (a)
+ ;; If the number is small enough, we don't need to use the full
+ ;; precision algorithm to compute the remainder. The value of 1024
+ ;; here is rather arbitrary. We should do an analysis to figure
+ ;; where the breakpoint should be.
+ (rem-pi/2-int a))
(defun sin-qd (a)
diff --git a/rt-tests.lisp b/rt-tests.lisp
index caf568f..69f6ee7 100644
--- a/rt-tests.lisp
+++ b/rt-tests.lisp
@@ -187,6 +187,18 @@
(check-accuracy 212 val true))
nil)
+;;; sin
+(rt:deftest oct.sin.pi
+ (not (zerop (sin +pi+)))
+ t)
+
+;;; cos
+(rt:deftest oct.cos.big
+ (let* ((val (cos (scale-float #q1 120)))
+ (err (abs (- val -0.9258790228548379d0))))
+ (<= err 5.2d-17))
+ t)
+
;;; Tests of atan where we know the analytical result
(rt:deftest oct.atan.1
(let* ((arg (/ (sqrt #q3)))
@@ -1731,6 +1743,6 @@
nil)
(rt:deftest erfc
- (check-accuracy 210 (erfc #q-4)
+ (check-accuracy 198 (erfc #q-4)
#q1.9999999845827420997199811478403265131159514278547464108088316570950057869589732)
nil)
-----------------------------------------------------------------------
Summary of changes:
qd-fun.lisp | 73 ++++++++++++++++++++++++++++++++++-----------------------
rt-tests.lisp | 14 ++++++++++-
2 files changed, 57 insertions(+), 30 deletions(-)
hooks/post-receive
--
OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. bcb0aded33f57a2ea901fdcc43fdfd88e8dff17b
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
This is an automated email from the git hooks/post-receive script. It was
generated because a ref change was pushed to the repository containing
the project "OCT: A portable Lisp implementation for quad-double precision floats".
The branch, master has been updated
via bcb0aded33f57a2ea901fdcc43fdfd88e8dff17b (commit)
from 0a607f3a6a6297597ae2a6c9896844803d42a6d9 (commit)
Those revisions listed above that are new to this repository have
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- Log -----------------------------------------------------------------
commit bcb0aded33f57a2ea901fdcc43fdfd88e8dff17b
Author: Raymond Toy <toy.raymond(a)gmail.com>
Date: Tue Nov 26 11:54:59 2013 -0800
Update the versions.
diff --git a/oct.asd b/oct.asd
index 1611fa8..aa430ae 100644
--- a/oct.asd
+++ b/oct.asd
@@ -36,7 +36,7 @@
:author "Raymond Toy"
:maintainer "See <http://www.common-lisp.net/project/oct>"
:licence "MIT"
- :version "2013.11.24" ; Just use the date
+ :version "2013.11.26" ; Just use the date
:components
((:file "qd-package")
(:file "qd-rep" :depends-on ("qd-package"))
@@ -76,7 +76,7 @@
(asdf:defsystem oct-tests
:depends-on (oct)
- :version "2011.12.05" ; Just use the date
+ :version "2013.11.26" ; Just use the date
:in-order-to ((compile-op (load-op :rt))
(test-op (load-op :rt :oct)))
:components
-----------------------------------------------------------------------
Summary of changes:
oct.asd | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
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OCT: A portable Lisp implementation for quad-double precision floats
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[oct-git]OCT: A portable Lisp implementation for quad-double precision floats branch master updated. 3c136f7ca5106ab99c95ffbc18a2b8c70d007342
by rtoy＠common-lisp.net 08 Apr '15
by rtoy＠common-lisp.net 08 Apr '15
08 Apr '15
This is an automated email from the git hooks/post-receive script. It was
generated because a ref change was pushed to the repository containing
the project "OCT: A portable Lisp implementation for quad-double precision floats".
The branch, master has been updated
via 3c136f7ca5106ab99c95ffbc18a2b8c70d007342 (commit)
from 6e6d810e0e1e9064dba180f758cad56a31877f1f (commit)
Those revisions listed above that are new to this repository have
not appeared on any other notification email; so we list those
revisions in full, below.
- Log -----------------------------------------------------------------
commit 3c136f7ca5106ab99c95ffbc18a2b8c70d007342
Author: Raymond Toy <rtoy(a)google.com>
Date: Sun Nov 24 20:53:22 2013 -0800
Clean up rem-pi/2.
o Remove old version
o Update new version to skip reduction if the arg <= pi/4.
diff --git a/qd-fun.lisp b/qd-fun.lisp
index 91c0b32..b3ae7b6 100644
--- a/qd-fun.lisp
+++ b/qd-fun.lisp
@@ -783,23 +783,12 @@ is the cosine of A"
(sub-qd f +qd-pi/2+))))))
(defun rem-pi/2 (a)
- ;; If the number is small enough, we don't need to use the full
- ;; precision algorithm to compute the remainder. The value of 1024
- ;; here is rather arbitrary. We should do an analysis to figure
- ;; where the breakpoint should be.
- (cond ((<= (abs (qd-0 a)) 256)
- (let ((quot (truncate (qd-0 (nint-qd (div-qd a +qd-pi/2+))))))
- (values (mod quot 4)
- (sub-qd a (mul-qd-d +qd-pi/2+ (float quot 1d0))))))
+ "Compute qd rem pi/2 = k + f, where k is an integer and |f| <
+ pi/4. Two values are returned: k mod 4 and f."
+ (cond ((qd-<= (abs-qd a) +qd-pi/4+)
+ (values 0 a))
(t
(rem-pi/2-int a))))
-
-(defun rem-pi/2 (a)
- ;; If the number is small enough, we don't need to use the full
- ;; precision algorithm to compute the remainder. The value of 1024
- ;; here is rather arbitrary. We should do an analysis to figure
- ;; where the breakpoint should be.
- (rem-pi/2-int a))
(defun sin-qd (a)
-----------------------------------------------------------------------
Summary of changes:
qd-fun.lisp | 19 ++++---------------
1 file changed, 4 insertions(+), 15 deletions(-)
hooks/post-receive
--
OCT: A portable Lisp implementation for quad-double precision floats
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